Overtone NMR is an experiment introduced by LeGros, Bloom, Tycko, and Opella, capable of providing N-14 powder spectra devoid of first-order quadrupole broadenings by irradiation and observation of the nuclear spins at twice their Larmor frequency. This technique constitutes one of the most promising alternatives for the acquisition of high resolution solid N-14 NMR spectra from random powders, particularly if it can be combined with strategies capable of removing the substantial second-order quadrupole broadenings remaining in the overtone line shapes. In order to facilitate the search for these averaging manipulations, we present here a theoretical description of the overtone experiment based on the time-domain propagation of density matrices. It is shown that by combining perturbation methods with appropriate rotating-frame transformations and diagonalizations, overtone spin-1 phenomena can be described using a single set of fictitious spin-1/2 operators. By contrast to conventional spin-1/2 irradiation and detection processes, however, overtone manipulations involve an unusual angular dependence on the azimuthal angle defining rotations about the main Zeeman magnetic field. This behavior introduces unexpected complications toward the narrowing of overtone resonances by conventional sample spinning techniques. Nevertheless, it can still be shown that the removal of all spin-1 anisotropies by certain forms of dynamic-angle spinning overtone NMR remains feasible.